Question

Given the sequence `{ 6,18,54,162,... }`, find the next three terms and an expression for the `n^(th)` term?

Answer 1

The relationship is that the `n^{th)` term is given by (assuming we start the sequence at `n=1`:

`u_n = 6xx3^(n-1)`

the next three terms are:

`486, 1458 ,4374`

Explanation:

The sequence:

`{ 6,18,54,162,... }`

clearly is not linear or quadratic as the terms increase too rapidly, we note that they are all even and factors of `6`, so we let us see how factoring out `6` helps to establish the pattern:

`{ 6,6xx3,6xx9,6xx27,... }`

and we can now see that the factor of `6` is multiplied by a power of `3`, giving us:

`{ 6,6xx3^1,6xx3^2,6xx3^3,... }`

And we also now that `3^0 = 1`, so we can modify the first term as follows:

`{ 6xx3^0,6xx3^1,6xx3^2,6xx3^3,... }`

And then we can see that the pattern is established and that the `n^{th)` term is given by (assuming we start the sequence at `n=1`:

`u_n = 6xx3^(n-1)`

So let us check that this works, and then form the next three terms:

`n=1 => u_1=6xx3^0 = 6`
`n=2 => u_2=6xx3^1 = 18`
`n=3 => u_3=6xx3^2 = 54`
`n=4 => u_4=6xx3^3 = 162`

`n=5 => u_5=6xx3^4 = 486`
`n=6 => u_6=6xx3^5 = 1458`
`n=7 => u_7=6xx3^6 = 4374`